Let k≥3 be an integer, Hkfalse(Gfalse) be the set of vertices of degree at least 2k in a graph G, and Lkfalse(Gfalse) be the set of vertices of degree at most 2k−2 in G. In 1963, Dirac and Erdős proved that G contains k (vertex) disjoint cycles whenever false|Hk(G)false|−false|Lk(G)false|≥k2+2k−4. The main result of this article is that for k≥2, every graph G with |V(G)|≥3k containing at most t disjoint triangles and with false|Hk(G)false|−false|Lk(G)false|≥2k+t contains k disjoint cycles. This yields that if k≥2 and false|Hk(G)false|−false|Lk(G)false|≥3k, then G contains k disjoint cycles. This generalizes the Corrádi–Hajnal Theorem, which states that every graph G with Hkfalse(Gfalse)=Vfalse(Gfalse) and false|Hk(G)false|≥3k contains k disjoint cycles.

We consider the following generalization of graph packing. Let G 1 = (V 1 , E 1 ) and G 2 = (V 2 , E 2 ) be graphs of order n and GWe extend the classical results of Sauer and Spencer and Bollobás and Eldridge on packing of graphs with small sizes or maximum degrees to the setting of list packing. In particular, we extend the well-known Bollobás-Eldridge Theorem, proving that if ∆(G 1 ) ≤ n−2, ∆(G 2 ) ≤ n−2, ∆(G 3 ) ≤ n−1, and |E 1 |+|E 2 |+|E 3 | ≤ 2n−3, then either (G 1 , G 2 , G 3 ) packs or is one of 7 possible exceptions. Hopefully, the concept of list packing will help to solve some problems on ordinary graph packing, as the concept of list coloring did for ordinary coloring.Mathematics Subject Classification: 05C70, 05C35.

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Dedicated to Adrian Bondy on the occasion of his 70 th birthday.then G 1 and G 2 pack. In the same paper, he conjectured that if ∆(G 1 ), ∆(G 2 ) ≤ n − 2, then |E 1 | + |E 2 | + max{∆(G 1 ), ∆(G 2 )} ≤ 3n − 7 is sufficient for G 1 and G 2 to pack. We prove that, up to an additive constant,Żak's conjecture is correct. Namely, there is a constant C such that if ∆(G 1 ), ∆(G 2 ) ≤ n − 2 and |E 1 |+|E 2 |+max{∆(G 1 ), ∆(G 2 )} ≤ 3n−C, then G 1 and G 2 pack. In order to facilitate induction, we prove a stronger result on list packing.

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